Optical accelerometer

ABSTRACT

Methods and apparatus optically measure acceleration, without Sagnac-effect corruption, without requiring slow light and without moving parts. Each optical accelerometer includes at least one measurement cell and at least one reference cell. Two optical signals traverse the cells in opposite directions around a figure-8-configured optical path and then interfere to produce an output signal. The reference cells have different indices of refraction than the measurement cells. Acceleration differentially affects speeds of the optical signals traversing the measurement and reference cells through differentially affecting the indices of refraction of the measurement and reference cells. These differences are evident in changes in the interference in the output signal, thereby enabling measurement of the acceleration. Several embodiments, including optical bench, vertical slab multi-pass, toroidal prism, planar waveguide, cylindrical waveguide, wound waveguide and optical fiber, are described.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation application of U.S. patentapplication Ser. No. 15/990,761 filed May 28, 2018, which is adivisional application of U.S. patent application Ser. No. 15/424,171filed Feb. 3, 2017, which claims the benefit of U.S. ProvisionalApplication No. 62/291,037, filed Feb. 4, 2016, each of which is herebyincorporated by reference in its entirety.

BACKGROUND

Conventional accelerometers contain moving parts and detect specificforces to ascertain acceleration. For example, a pendulous integratinggyroscopic accelerometer (PIGA) includes a spinning gyroscope and apendulous mass mounted on a bearing. A spring-mass accelerometer employsa proof mass suspended within a frame by one or more springs andmeasures displacement of the proof mass, relative to the frame, as aresult of acceleration. Moving parts make such conventionalaccelerometers difficult to manufacture and subject to mechanical wear.

U.S. Pat. No. 7,324,205 to Robert J. Howard (“Howard”) discloses adevice, said to be an optical accelerometer, gravitometer andgradiometer. However, Howard's device requires “slow light,” i.e., lightthat travels at significantly less than c, the speed of light in avacuum. Slow light may result from light passing through a medium thathas been cooled to near absolute zero to produce a Bose-Einsteincondensate (BEC), which causes electromagnetically-induced transparency(EIT) of an ordinarily opaque material. Howard's device requirescumbersome apparatus to produce slow light. Furthermore, Howard's deviceis subject to Sagnac-effect corruption.

In his 1868 paper “Determination de la vitesse avec laquelle estentrainée une onde lumineuse traversant un milieu en movement”(“Determination of the Speed with which a Light Wave is EntrainedCrossing a Moving Medium”), Martin (also Martinus) Hoek (“Hoek”)disclosed an unbalanced interferometer.

U.S. Pat. No. 6,813,006 to Ruyong Wang, et al. (“Wang”) improves on theHoek device and discloses an optical device said to measure speed of anobject by detecting interference between two electromagnetic beams thatpass through two media having different dispersive dragging effects onthe beams. However, Wang's description of his device as a “velocimeter”is believed to be incorrect, as his device is not sensitive to velocity.Instead, Wang's device is sensitive to acceleration, i.e., change invelocity between successive inertial velocity states.

SUMMARY OF EMBODIMENTS

An embodiment of the present invention provides an opticalaccelerometer. The optical accelerometer has an input measurement axis.

The optical accelerometer includes first and second counterfacingelongated retroreflectors. The retroreflectors have parallel respectivelength axes. The retroreflectors define a first optical circuit betweenthe first and second retroreflectors. The first optical circuit bounds afirst area, as seen in a top view. The first optical circuit includesparallel first and second legs. The first and second legs extend betweenthe first and second retroreflectors.

The optical accelerometer also includes a beam splitter. The beamsplitter is positioned to divide an optical beam into first and secondbeams. The beam splitter is also positioned to introduce the first andsecond beams into the first optical circuit at respective first andsecond angles to the length axis of the first retroreflector. The firstand second angles are other than perpendicular. The first and secondbeams traverse the first optical circuit in opposite directions, as seenin the top view. The first and second beams traverse the first opticalcircuit along respective rectangular helical paths. Each helical pathincludes a plurality of loops.

The optical accelerometer also includes third and fourth counterfacingelongated retroreflectors. The third and fourth retroreflectors haverespective length axes. The length axes of the third and fourthretroreflectors are parallel to the length axis of the firstretroreflector. The third and fourth retroreflectors define a secondoptical circuit between the third and fourth retroreflectors. The secondoptical circuit bounds a second area, as seen in a top view. The secondarea is equal in size to the first area. The second optical circuitincludes third and fourth legs. The third and fourth legs extend betweenthe third and fourth retroreflectors. The third and fourth legs extendparallel to the first leg.

The optical accelerometer also includes a first stack coupling mirror.The first stack coupling mirror is disposed in the first leg to reflectthe first beam toward and into the second optical circuit. The firstbeam traverses the first and second optical circuits in oppositedirections, as seen in the top view.

The optical accelerometer also includes a second stack coupling mirror.The second stack coupling mirror is disposed in the second leg toreflect the second beam toward and into the second optical circuit. Thesecond beam traverses the first and second optical circuits in oppositedirections, as seen in the top view.

The optical accelerometer also includes first and second optical media.The first and second optical media have substantially equal refractiveindices. The first and second optical media have substantially equallengths. The first and second optical media are disposed, respectively,in the first and second optical circuits. The first beam traverses bothoptical media in equal directions. The first beam traverses both opticalmedia parallel to the input measurement axis. The first and second beamstraverse each of the optical media in opposite directions, as seen inthe top view.

The optical accelerometer also includes third and fourth optical media.The third and fourth optical media have substantially equal refractiveindices. The refractive indices of the third and fourth optical mediaare different from the refractive indices of the first and secondoptical media. The third and fourth optical media are disposed,respectively, in the first and second optical circuits. The first beamtraverses the first optical medium and the third optical medium inopposite directions. The first beam traverses the second optical mediumand the fourth optical medium in opposite directions.

The optical accelerometer also includes a beam combiner. The beamcombiner is positioned to combine the first and second beams. The beamcombiner is positioned to generate an output signal from interferencebetween the first and second beams, after the first and second beamshave traversed the plurality of loops.

The refractive index of each of the first and second optical media maybe at least about 1.3. The refractive index of each of the third andfourth optical media may be at most about 1.002.

The first optical medium may include glass. The second optical mediummay include glass. The third optical medium may include a first chambercontaining a medium. The medium may be air, vacuum or at least 99%nitrogen. The fourth optical medium may include a second chambercontaining a medium as selected for the first chamber.

The first optical medium may include a first right rectangular prism.The second optical medium may include a second right rectangular prism.

The beam combiner may be optically coupled to the second optical circuitat a far point, along the first and second optical circuits, from thebeam splitter.

The optical accelerometer may also include a first reversing mirror. Thefirst reversing mirror may be disposed in the second optical circuit atan angle complementary to the first angle, relative to the length axisof the first retroreflector, to reflect the first beam. A secondreversing mirror may be disposed in the second optical circuit at anangle complementary to the second angle, relative to the length axis ofthe first retroreflector, to reflect the second beam. The beam splittermay include the beam combiner.

Another embodiment of the present invention provides an opticalaccelerometer. The optical accelerometer has an input measurement axis.

The optical accelerometer includes first and second counterfacingtoroidal retroreflectors. The first and second counterfacing toroidalretroreflectors have their toroidal centers disposed on the inputmeasurement axis.

The optical accelerometer also includes a first toroidal optical medium.The first toroidal optical medium has a first refractive index. Thefirst toroidal optical medium is disposed between the first and secondcounterfacing toroidal retroreflectors. The first toroidal opticalmedium is disposed coaxial with the input measurement axis.

The optical accelerometer also includes a second toroidal opticalmedium. The second toroidal optical medium has a second refractiveindex. The second refractive index is different from the firstrefractive index. The second refractive index is disposed between thefirst and second counterfacing toroidal retroreflectors. The secondrefractive index is disposed coaxial with the input measurement axis.

The first and second counterfacing toroidal retroreflectors and thefirst and second optical media define a first optical path extendingfrom the first retroreflector, through the first optical medium, to thesecond retroreflector. The first and second counterfacing toroidalretroreflectors and the first and second optical media define a secondparallel optical path from the first retroreflector, through the secondoptical medium, to the second retroreflector. The first optical path andthe second parallel optical path have substantially equal lengths.

The optical accelerometer also includes a beam splitter. The beamsplitter is positioned to divide an optical beam into first and secondbeams. The beam splitter is positioned to introduce the first and secondbeams, respectively, into the first and second optical paths notparallel to the input measurement axis. Consequently, the first andsecond beams alternatingly reflect from the first and secondcounterfacing toroidal retroreflectors at progressive circumferentiallocations along each of the first and second retroreflectors. The firstand second beams alternatingly traverse the first and second opticalmedia. The first and second beams traverse the first optical medium inopposite directions.

The optical accelerometer also includes a beam combiner. The beamcombiner is positioned to combine the first and second beams. The beamcombiner is positioned to generate an output signal from interferencebetween the first and second beams after the first and second beams havetraversed each of the first and second optical media a plurality oftimes.

The first refractive index may be at least about 1.3. The secondrefractive index may be at most about 1.002.

The optical accelerometer may also include a reversing mirror.

The beam splitter may include the beam combiner.

The optical accelerometer may also include an electronic circuit. Theelectronic circuit may be disposed within a minor diameter of the secondtoroidal optical medium. The electronic circuit may be coupled toreceive the output signal. The electronic circuit may be coupled togenerate an acceleration signal based on the output signal.

Yet another embodiment of the present invention provides a method formanufacturing optical fiber. The method includes providing zero-areaSagnac rotation compensation. The method further includes irradiatingthe optical fiber intermittently along its length with one or aplurality of irradiation wavelengths. In an embodiment of this method,the step of providing a zero-area Sagnac rotation compensation includeswinding the optical fiber in a first winding direction; and includeswinding the optical fiber in a second winding direction, to form firstand second half-winding pairs, whereby the first and second half-windingpairs are capable of conducting a light beam in first and seconddirections, respectively, and opposite each other, relative to an inputaxis. In an embodiment of this method, the step of irradiating theoptical fiber includes irradiating with a plurality of irradiationintensities. In an embodiment of this method, the step of intermittentlyirradiating the optical fiber includes using one or more lasers. In anembodiment of this method, the optical fiber comprises fiber with aplurality of indices of refraction along its length. In an embodiment ofthis method, the optical fiber comprises at least one double-cladlong-period fiber Bragg grating.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be more fully understood by referring to thefollowing Detailed Description of Specific Embodiments in conjunctionwith the Drawings, of which:

FIGS. 1 and 2 are respective top and side schematic diagrams of anoptical bench optical accelerometer interferometer arrangement,according to an embodiment of the present invention.

FIG. 3 is an exploded perspective-view schematic diagram of one elementof a vertical slab multi-pass optical accelerometer, according to anembodiment of the present invention.

FIGS. 4, 5 and 8 are cross-sectional schematic diagrams of the elementof FIG. 3.

FIGS. 6 and 7 are perspective-view and side schematic diagrams,respectively, of a body of the element of FIG. 3.

FIGS. 9, 10 and 11 are cross-sectional schematic diagrams of twoside-by-side elements of FIG. 3 forming the vertical slab multi-passoptical accelerometer.

FIG. 12 is a perspective schematic diagram of the vertical slabmulti-pass optical accelerometer of FIGS. 9-11.

FIG. 13 is a cut-away perspective schematic diagram of a toroidaloptical accelerometer, according to another embodiment of the presentinvention.

FIG. 14 is a cut-away portion of the toroidal optical accelerometer ofFIG. 13 illustrating retroreflectors thereof, according to an embodimentof the present invention.

FIG. 15 is a perspective schematic diagram of an element of the toroidaloptical accelerometer of FIG. 13, curved to wrap it about an input axis.

FIG. 16 is another cut-away view of the toroidal optical accelerometerof FIGS. 13-15, and FIG. 17 is a cross-sectional view of the toroidaloptical accelerometer taken through a beam splitter thereof.

FIGS. 18 and 19 are cut-away view of the toroidal optical accelerometer,similar to FIGS. 16 and 17, schematically illustrating paths taken byportions of two beams, according to an embodiment of the presentinvention.

FIGS. 20 and 21 are respective plan and left side schematic diagrams ofa planar waveguide optical accelerometer, according to anotherembodiment of the present invention.

FIG. 22 includes a plan view of a portion of the planar waveguideoptical accelerometer of FIGS. 20 and 21, according to an alternativeembodiment of the present invention.

FIG. 23 is a perspective schematic illustration of a cylindricalwaveguide optical accelerometer, according to an embodiment of thepresent invention.

FIGS. 24 and 25 are perspective schematic illustrations of a nestedcylindrical waveguide optical accelerometer that includes severalcoaxial cylinders, each similar to the cylinder of FIG. 24, according toanother embodiment of the present invention.

FIG. 26 is a perspective schematic illustration of a spiral-woundwaveguide optical accelerometer 2600, according to yet anotherembodiment of the present invention.

FIG. 27 is a perspective schematic illustration of a fiber opticaccelerometer according to another embodiment of the present invention.

FIG. 28 schematically illustrates a same portion of an optical fiber, aswell as schematically illustrating four cross-sections of the opticalfiber, of the fiber optic accelerometer of FIG. 27.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

FIGS. 1 and 2 are respective top and side schematic diagrams of anoptical bench optical accelerometer 100, arranged according to anembodiment of the present invention. The optical accelerometer 100includes a special case interferometer. Embodiments of the presentinvention operate according to principles that will be explained usingFIGS. 1 and 2. The optical accelerometer 100 includes a light source105. The light source 105 can emit any suitable electromagneticradiation, such as visible or invisible, such as infrared, light. Forexample, the light source 105 may be a laser that emits light having awavelength of about 1,550 nm. A light beam 110 emitted by the lightsource 105 enters a first beam splitter 115, where the beam 110 is splitinto a first beam 120 and a second beam 125.

The first beam 120 is reflected by a first mirror 130. The first beam120 then traverses a first optical medium 135. The first optical medium135 has an index of refraction (N1) greater than 1. For example, thefirst optical medium 135 may include a high-quality optical glass havingan index of refraction of about 1.3, 1.4 or 1.46. Other optical mediahaving other indices of refraction are acceptable.

The second beam 125 traverses a third optical medium 140, such as apartial or complete vacuum, air or dry nitrogen, having an index ofrefraction (N3) different from the index of refraction of the firstoptical medium 135 (N1). The third optical medium 140 may be confinedwithin a sealed chamber. In the example depicted in FIG. 1, the index ofrefraction of the third optical medium 140 is about 1. Gain of theoptical accelerometer 100 depends in part on a difference between theindex of refraction of the first optical medium 135 and the index ofrefraction of the third optical medium 140, with a larger differenceproviding greater gain.

A path taken by the first beam 120 through the first optical medium 135should be at least as long 145 as a path taken by the second beam 125through the third optical medium 140. In the embodiment shown in FIG. 1,the paths taken through the first and third optical media 135 and 140,respectively, are substantially equal in length. As used herein,“substantially equal” lengths means the lengths do not differ by morethan a relatively small fraction of a wavelength of the first and secondbeams 120 and 125, such that any such difference does not contribute tomore than about a few percent error.

After exiting the first optical medium 135, the first beam 120 traversesa second optical medium 150. The second optical medium 150 has an indexof refraction (N2) greater than 1. The index of refraction of the secondoptical medium 150 (N2) can, but need not, be the same as the index ofrefraction of the first optical medium 135 (N1). The second opticalmedium 150 can, but need not, be made of the same material as the firstoptical medium 135. The second optical medium 150 may include ahigh-quality optical glass having an index of refraction of about 1.3,1.4 or 1.46. Other optical media having other indices of refraction areacceptable.

The first beam traverses both the first and the second optical media 135and 150 in equal directions. “Equal directions” means the directions areparallel, although not necessarily collinear, and not opposite. As usedherein, “equal directions” has a meaning similar to equal directions, asused to refer to equal direction vectors. In the example of FIG. 1, thebeams through the first and second optical media 135 and 150 are alsocollinear.

After exiting the third optical medium 140, the second beam 125 isreflected by a second mirror 155 and a third mirror 157, and then thesecond beam 125 traverses a fourth optical medium 160. The fourthoptical medium 160 has an index of refraction different from the indexof refraction of the second optical medium 150. The fourth opticalmedium 160 can, but need not, be the same material as in the thirdoptical medium 140. In the example depicted in FIG. 1, the index ofrefraction of the fourth optical medium 160 is about 1. The second beam125 traverses the third and fourth optical media 140 and 160 in equaldirections. Gain of the optical accelerometer 100 depends in part on adifference between the index of refraction of the second optical medium150 and the index of refraction of the fourth optical medium 160, with alarger difference providing greater gain.

A path taken by the first beam 120 through the second optical medium 150should be at least as long as a path taken by the second beam 125through the fourth optical medium 160. In the embodiment shown in FIG.1, the paths taken through the second and fourth optical media 150 and160, respectively, are substantially equal in length.

Collectively, fourth and fifth mirrors 165 and 170 couple the pathstaken thus far by the first and second beams 120 and 125 to each other,so the first beam 120 now follows the path of the second beam 125, butin reverse, and the second beam 125 now follows the path of the firstbeam 120, but in reverse, back to the beam splitter 115.

In a simplified version of the optical accelerometer 100 shown in FIG.1, the beam splitter 115 combines the first and second beams 120 and 125to provide an output signal 175. A difference in path lengths followedby the first and second beams 120 and 125 produces interference when thebeam splitter 115 combines the two beams 120 and 125. This interferencecan be measured by a suitable sensor 180, such as an optical sensor, ina Sagnac configuration.

Alternatively, one of the mirrors 165 or 170 may be replaced by a beamcombiner 195, shown in an insert in FIG. 1, in which case an outputsignal 175 a is provided by the beam combiner 195 to a sensor 180 a in aMach-Zehnder configuration.

The overall path lengths of the first and second beams 120 and 125 fromthe beam splitter 115, through the optical media 135, 140, 150 and 160,and back to the beam splitter 115 (or alternatively to the beam combiner195) should be equal. As a practical matter, it may be impossible toinitially position the above-described components of the opticalaccelerometer 100 such that the path lengths are identical. Therefore,the light source 105, the beam splitter 115 and the sensor 180 may bemounted on a table 185 that is moveable, relative to the othercomponents, and a zero-adjustment screw 190 may be used to adjust theposition of the table 185 vertically (as viewed in FIG. 1) to compensatefor the path length difference. Alternately, any apparent path lengthdifference with no acceleration input may be compensated for byadjusting electronics subsequent to the sensor 180 or 180 a.

As noted, FIG. 2 is a side view of the optical accelerometer 100 ofFIG. 1. The boresight axis of the light source 105 is tilted at anangle, other than zero relative to horizontal. Therefore, the light beam110 enters the first beam splitter 115 at an angle, and the first andsecond beams 120 and 125 travel at angles until they reach the mirrors165 and 170. One or both of the mirrors 165 and 170 are tilted by anangle, so the return paths of the light beams 120 and 125 are alsotilted. Therefore, the first and second beams 120 and 125 travel alongessentially spiral paths, from the viewpoint of FIG. 2, up in theexample of FIG. 2, until they are combined by the beam splitter 115 (orby the combiner 195 of the insert of FIG. 1). The boresight axis of thesensor 180 or 180 a is tilted at an angle opposite to the angle of thelight source 105.

The first beam 120 travels from the first beam splitter 115 to the firstmirror 130, through second mirrors 157 and 155 and through the thirdoptical medium 140 in a generally clockwise direction, as indicated byan arrow 196. These four legs collectively comprise a first opticalcircuit 191. After passing through optical medium 135, the first beam120 continues through the second optical medium 150, the mirrors 165 and170, the fourth optical medium 160 and completes the upper half of thepath between the third and second mirrors 157 and 155 in a generallycounterclockwise direction, as indicated by an arrow 197. These fourlegs collectively comprise a second optical circuit 192.

The first optical circuit 191 bounds an area 193, as seen in the topview of FIG. 1. The second optical circuit 192 bounds an area 194, asseen in top view of FIG. 1. In a perfectly constructed or adjusteddevice, the area 193 bounded by the first optical circuit is equal tothe area 194 bounded by the second optical circuit, the path length ofthe first optical circuit 191 is equal to the path length of the secondoptical circuit 102 and the first beam 120 traverses the first andsecond optical circuits 191 and 192, respectively, in clockwise andcounterclockwise directions. Similarly, the second beam 125 traversesthe first and second optical circuits 191 and 192, respectively, incounterclockwise and clockwise directions. The first and second beams120 and 125 each traverse each of the respective optical circuits 191and 192 in opposite directions. The two optical circuits 191 and 192effectively form a figure-8 configuration for each of the beams 120 and125. This arrangement cancels any Sagnac effect on the beams 120 and 125due to rotation of the device 100 in the inertial frame.

The optical media 135, 140, 150 and 160 are oriented parallel to aninput measurement axis 198. The direction of travel of the first beam120 through the first and second optical media 135 and 150 is in thesame direction as, or at least parallel to, the input measurement axis198. Similarly, the direction of travel of the second beam 125 throughthe third and fourth optical media 140 and 160 is in the same directionas, or at least parallel to, the input measurement axis 198. The firstand second optical media 135 and 150 are referred to herein as“measurement cells,” and the third and fourth optical media 140 and 160are referred to herein as “reference cells.” The first optical circuit191 includes a measurement cell and a reference call, and the secondoptical circuit 192 includes a measurement cell and a reference cell.

In operation, the optical accelerometer 100 is initially placed in alocal inertial frame with V₀ equal to a constant (e.g., measurement axis198 is aligned horizontally, and alternately East and West) and the pathlength of one optical circuit is adjusted by the zero-adjustment screw190, so that the resultant interference pattern measured by the sensor180 or 180 a is nulled. Although the effect on Sagnac-effectcancellation is minimal when initially zeroing the interference pattern,additional zeroing procedures, zero-adjustment mechanisms or electricalcompensation circuits may be added as desired to further improveaccuracy, as is common practice in the inertial component art. When thedevice is accelerated, so that the velocity changes (V₁≠V₀), theincrease in kinetic energy of the device 100 causes an unequal change inthe refractive indices of the measurement and reference cells (opticalmedia 135, 150, 140 and 160, respectively). If the medium in thereference cells (optical media 140 and 160) is vacuum, their refractiveindices remain constant (n_(r)=1) and unaffected by the change invelocity of the device 100. During acceleration between V₀ and V₁, thefrequencies of the first beam 120 passing through the measurement cells135 and 150 and of the second beam 125 passing through the referencecells 140 and 160 changes, one increasing, and the other decreasing. Theresultant beat frequency between them is a measure of the accelerationalong the measurement axis 198, a=(dV/dt)·(IA), where an underlineindicates a vector quantity, and the dot indicates a vector dot product.While the acceleration continues, the beat frequency integrates into aphase shift between beams 120 and 125. After the acceleration stops, thechange in the interference pattern due to the integrated phase changeduring acceleration is a permanent measure of the change in velocity(integrated acceleration), ΔV=(V₁−V₀)·(IA) since nulling at V₀ (launchtime).

The Fizeau Effect (Hippolyte Fizeau, 1851) is a demonstration that thespeed of light in a moving medium is different from the speed of lightin a stationary medium, although Fizeau's conclusion (that theluminiferous aether was dragged by the moving medium, thereby verifyingexistence of the aether) is incorrect. The optical accelerometer 100directly detects a change in velocity by a shift in fringes of theinterference pattern, as seen by the detector 180 or 180 a. The shift inthe fringes results from the first and second beams 120 and 125 beingdifferently affected by the change in velocity, as they pass through theoptical media 135/150 and 140/160, which have different indices ofrefraction. The effect is due to acceleration of a dielectric (the firstand second optical media 135 and 150) in space-time affectinginterchange of energy between photons of the light beams and atoms ofthe dielectric medium, effectively changing the dielectric constant inthe direction of acceleration 198.

Photons passing through a transparent dielectric medium mostly travelthrough vacuum between atoms, with occasional interaction with the atomsof the dielectric. In a constant velocity inertial frame, the momentumexchange with the atoms of the dielectric gives rise to phase and phasevelocity changes characterized by the parameters known as dielectricconstant and index of refraction of the material. An acceleratingtransparent dielectric transfers additional momentum to theelectromagnetic field of the photon wave packet over that which istransferred at constant velocity. This changes the photon's phase andfrequency from what it would be in a non-accelerating inertial frame.The integrated effect over the photon beam's path through the medium isa change in its instantaneous phase velocity through the dielectric,which is equivalent to a change in the index of refraction of themedium.

When the acceleration stops, the beam frequency within the mediumreturns to its original value, but the integrated change in frequencyremains as a permanent change in phase and is a measure of theintegrated acceleration, ΔV, from the (arbitrary) steady-state velocityof the original inertial frame to the new steady-state velocity(arbitrary+ΔV) of the new inertial frame.

The optical media 140 and 160 have indices of refraction equal, orclose, to 1 and are not, therefore, affected by any acceleration or areaffected to only a small degree. The vacuum wavelength, initial lengthof the measurement cell and measured phase difference under accelerationare all scalars and invariant to Lorenz transformation back to thelaunch coordinate system within which the vehicle trajectory iscalculated. The system is ratiometric and does not requiresynchronization of any clocks. The effect of using air or anotherdielectric of small index for the reference cell instead of vacuumdecreases the magnitude of the effect, i.e., scale factor, but notsignificantly.

Acceleration measured by the optical accelerometer 100 is over and aboveany true Doppler frequency effects due to a differential velocitybetween the photon source 105 and detector 180 or 180 a during thepassage of the photons during the acceleration period. These, and anyrelated frequency shifts at interfaces between areas of differentindices of refraction, cancel out in a properly designed interferometer.The moving system clock frequency also changes by a time dilation of thewhole process, but this is taken into account by the Lorenztransformations of the photon transit velocity.

The device 100 can operate in either a frequency-tracking accelerometermode, analogous to an interferometric fiber-optic gyroscope (IFOG), orin an integrated Δ-velocity mode, analogous to an integrating gyroscopicaccelerometer (PIGA). If adequate separation is provided betweenadjacent narrow beams on successive passes around the loops, which arein a figure-8 configuration (“figure-8 loops”), in the same direction toavoid mode locking, such as sufficiently tilting the plane of the beamsas in FIG. 2, the device 100 operates in the integrated Δ-velocity mode,and successive passes provide increased effective path length andsensitivity. Alternately, if a wide beam floods the measurement andreference cells 135, 150, 140 and 160, the device 100 operates in thefrequency-tracking accelerometer mode, as an etalon, with interferencebetween successive passes narrowing the output beam bandwidth,increasing the ability to discriminate differences in frequency, butwith no increase in effective path length on successive passes. In thiscase, Δ-velocity integration may most easily be accomplished ininstrument electronics, as in an IFOG. Mode locking betweencounter-propagating beams may need to be handled as in IFOGs. Dynamicpolarization effects may need to be taken into account as well. Thelight source 105 and sensor 180 or 180 a may be conventionallyfabricated, as for an IFOG, such as in an integrated optical chip (IOC)using serrodyne modulation. A low coefficient of thermal expansionmaterial, such as Zerodur® glass, available from Schott AG, Mainz,Germany, may be used to fabricate components and/or a base for thecomponents of the optical accelerometer 100.

FIG. 3 is an exploded perspective schematic diagram of one element 300of a vertical slab multi-pass optical accelerometer, according to anembodiment of the present invention. The vertical slab multi-passoptical accelerometer includes a special case interferometer. Thevertical slab multi-pass optical accelerometer includes two or more suchelements 300, although some of the elements are mirror images of theelement 300 depicted in FIG. 3, as described in more detail herein. Eachelement 300 may be made of any suitable material, such as a lowcoefficient of thermal expansion material, such as Zerodur® glass.

The element 300 includes a top 305, a bottom 310 and a body 315. Thebody 315 includes a first optical medium 320. The first optical medium320 may be in the form of a first right rectangular prism. The top,bottom and body 300-310 define a void 325 that may be filled with athird optical medium, such as air, dry nitrogen or a liquid, or the void325 may be partially or completely evacuated. The index of refraction ofthe first optical medium 320 is different from the index of refractionof the third optical medium. The first optical medium 320 forms ameasurement cell, and the third optical medium forms a reference cell,as described with respect to the embodiment of FIGS. 1-2.

FIG. 3 shows a section plane 330, and FIG. 4 is a cross-sectionalschematic diagram of the element 300, as viewed across the section plane330. The body 315 includes two mirrors 400 and 405, collectively forminga first elongated retroreflector 407, and two additional mirrors 410 and415 collectively forming a counterfacing second elongated retroreflector417. The retroreflectors 407 and 417 are elongated parallel to avertical axis 335, as shown in FIG. 3. The retroreflectors 407 and 417have length axes 340 and 345, respectively, parallel to each other andparallel to the vertical axis 335. As used herein, the term“retroreflector” means a pair of mirrors or another arrangement thatreflects light along an axis that is parallel, in at least twodimensions, to an incoming light axis. In some embodiments, aretroreflector includes a pair of planar mirrors, thereby permitting thelight to be reflected in a third dimension, as discussed in more detailwith reference to FIG. 7. In FIG. 7, the retroreflectors 407 and 417create a helical path 705 that “walks up” the body 315. In some cases,the retroreflectors described herein, may be implemented with Porromirrors or Porro prisms.

Although section lines in FIG. 4 indicate the retroreflectors 407 and417, the first optical medium 320 and sides 420 and 425 of the body 315are distinct materials or distinct parts, any combination of these partsmay be fabricated as a whole. FIG. 5 is another cross-sectional view ofthe body 315 providing the same view as in FIG. 4, except FIG. 5 omitsthe section lines for clarity.

The retroreflectors 407 and 417 are disposed, relative to the firstoptical medium 320 and the void 325, to define a first optical circuit500 between the first and second retroreflectors 407 and 417. The firstoptical circuit 500 includes legs 505, 510, 515 and 520. The first leg505 is parallel to second leg 515. These legs 505 and 515 extend betweenthe first and second retroreflectors 407 and 417. Leg 505 traverses thefirst optical medium 320, and leg 515 traverses the void 325, whichcontains the third optical medium. Collectively, the legs 505-520 bounda first area 525, as seen in the top view provided in FIGS. 4 and 5.

FIG. 6 is a another perspective schematic diagram of the body 315showing the outline of the void 325, faces of the retroreflectors 407and 417 and the first optical medium 320. Other lines of the body 315are omitted for clarity. A beam splitter 600 is disposed, relative tothe void 325, the faces of the retroreflectors 407 and 417 and the firstoptical medium 320.

A light source 605 emits a light beam 610 into the beam splitter 600.The beam splitter 600 splits the light beam 610 into a first beam 615and a second beam 620. The first beam 615 reflects from one surface ofthe retroreflector 407 and then enters the first optical medium 320. Thesecond beam 620 passes through the beam splitter 600 and enters thethird optical medium in the void 325.

FIG. 7 is a side schematic diagram of the body 315. The boresight axisof the light source 605 is tilted at an angle 700, other than zerorelative to horizontal. Therefore, the light beam 610 enters the beamsplitter 600 at an angle, and the first and second beams 615 and 620travel at angles, relative to horizontal. Only a path 700 of the firstbeam 615 is shown in FIG. 7 for clarity. When the first beam 615 reachesthe second retroreflector 417, the first beam 615 is reflected at anangle, relative to horizontal, equal to the angle 700. Similarly, whenthe first beam 615 then reaches the first retroreflector 407, it isagain reflected at an angle, relative to horizontal, equal to the angle700. Thus, the first beam 615 follows a rectangular (as seen in topview) helical path 705 up (as seen in side view) the element 300. FIG. 8is a top schematic view, similar to FIG. 5, but including the lightsource 605 and the beam splitter 600. The second beam 620 (not shown inFIG. 7) follows a similar rectangular (as seen in top view) helical pathup the element 300, although the second beam 620 traverses the firstoptical circuit 500 in a direction opposite the first beam 615.

Because the angle of the light source 605 is other than zero, relativeto horizontal, the angles of the light beams 615 and 620 are other thanperpendicular, relative to the length axis 340 of the firstretroreflector 407. Thus, the beam splitter 600 introduces the first andsecond beams 615 and 620 into first optical circuit 500 at respectivefirst and second angles, other than perpendicular, to the length axis345 of the first retroreflector 407. The first and second beams 615 and620 traverse the first optical circuit 500 in opposite directions, asseen in the top view (FIG. 8), along respective rectangular helicalpaths. Each helical path includes a plurality of loops as the first andsecond beams 615 and 620 climb the element 300 (FIG. 7).

FIG. 8 is a top schematic view, similar to FIG. 5, but including thebeam splitter 600 and light source 605. As can be seen in FIG. 8, thefirst beam 615 traverses the first optical path 500 in a generallycounterclockwise direction, as indicated by arrow 800. The two beams 615and 620 counter-rotate around the first optical circuit 500. That is,the first and second beams 615 and 620 traverse the first opticalcircuit 500 in opposite directions. Thus, the second beam 620 traversesthe first optical path 500 in a generally clockwise direction.

FIG. 9 is a top cross-sectional view of a vertical slab multi-passoptical accelerometer 900 that includes two side-by-side, largelymirror-image, elements 300 and 300′. Each of the elements 300 and 300′may be considered a “stack.” The two elements 300 and 300′ may share acommon wall 905. The second element 300′ includes third and fourthcounterfacing elongated retroreflectors 407′ and 417′, a second opticalmedium 320′ and it defines a second void 325′ filled with a fourthoptical medium. The second optical medium 320′ may be in the form of asecond right rectangular prism.

The third and fourth counterfacing elongated retroreflectors 407′ and417′ have respective length axes parallel to the length axis 340 (FIG.3) of the first retroreflector 407. The third and fourth counterfacingelongated retroreflectors 407′ and 417′ define a second optical circuit500′ between the third and fourth retroreflectors 407′ and 417′. Thesecond optical circuit 500′ bounds a second area 525′, as seen in a topview, equal in size to the first area 525 bounded by the first opticalcircuit 500. The second optical circuit 500′ includes a third leg 505′and a fourth leg 515′, each of which extends between the third andfourth retroreflectors 407′ and 417′.

FIG. 10 is a top cross-sectional view similar to FIG. 9, but takenhigher up the vertical slab multi-pass optical accelerometer 900. Abovethe top of the retroreflector 417 is a first stack coupling mirror 1000.The first stack coupling mirror 1000 is disposed in the first leg 505 toreflect the first beam 615 toward and into the second optical circuit500′. As a result, the first beam 615 traverses the first and secondoptical circuits 500 and 500′ in opposite directions, as seen in the topview and as indicated by arrows 800 and 800′, along respectiverectangular helical paths. Each helical path includes a plurality ofloops. The number of loops in the first optical circuit 500 should equalthe number of loops in the second optical circuit 500′. The first beam615 traverses the first and second optical media 320 and 320′ in equaldirections and parallel to an input measurement axis 1005.

Also above the top of the retroreflector 417 in the first stack is asecond stack coupling mirror 1010, and above the retroreflector 417′ inthe second stack is a third stack coupling mirror 1015. The second stackcoupling mirror 1010 is disposed in the second leg 515. Collectively,the second and third stack coupling mirrors 1010 and 1015 reflect thesecond beam 620 toward and into the second optical circuit 500′. As aresult, the second beam 620 traverses the first and second opticalcircuits 500 and 500′ in opposite directions, as seen in the top view.Nevertheless, the second beam 620 traverses the first and second opticalmedia 320 and 320′ in equal directions and anti-parallel to the inputmeasurement axis 1005.

The first and second optical media 320 and 320′ may have substantiallyequal refractive indices and substantially equal lengths. As usedherein, “substantially equal” refractive indices means the refractiveindices do not differ by more than a relatively small fraction, suchthat any such difference does not contribute to more than about a fewpercent error. The first and second optical media 320 and 320′ formmeasurement cells, as described with respect to the embodiment of FIGS.1-2. Similarly, the third and fourth optical media in the voids 325 and325′ may have substantially equal refractive indices different from therefractive indices of the first and second optical media 320 and 320′,and the third and fourth optical media in the voids 325 and 325′ formreference cells, as described with respect to the embodiment of FIGS.1-2. The optical media are disposed, respectively, in the first andsecond optical circuits 500 and 500′, such that the first beam 615traverses the first optical medium 320 and the third optical medium inopposite directions, and the first beam 615 traverses the second opticalmedium 325′ and the fourth optical medium in opposite directions.

The stack coupling mirrors 1000, 1010 and 1015 are tilted downward (inside view) to reverse the upward helical path of the beams 615 and 620.Thus, in the second element 300′, the first beam 615 follows arectangular (as seen in top view) helical path down (as seen in sideview) the second element 300′, analogous to the way the first beam 615follows a rectangular helical path up the first element 300, asdescribed with reference to FIG. 7. The second beam follows a similarrectangular (as seen in top view) helical path down the second element300′, although the second beam 620 traverses the second optical circuit500′ in a direction opposite the first beam 615.

FIG. 11 is a top cross-sectional view similar to FIG. 10, but takenlower down the vertical slab multi-pass optical accelerometer 900. Abeam combiner 1100 is disposed, relative to the void 325′, the faces ofthe retroreflectors 407′ and 417′ and the second optical medium 320′.The beam combiner 1100 combines the first and second beams 615 and 620to produce an output signal 1105 from interference between the first andsecond beams 615 and 620, after the first and second beams 615 and 620have traversed the plurality of loops in the first and second opticalcircuits 500 and 500′. A sensor 1110 receives the output signal 1105 todetect the interference. FIG. 12 is a perspective schematic diagram ofthe vertical slab multi-pass optical accelerometer 900.

As shown and described, the beam combiner 1100 is optically coupled tothe second optical circuit 500′ at a far point, along the first andsecond optical circuits 500 and 500′, from the beam splitter 600 (FIGS.6 and 8) in a Mach-Zehnder configuration. Alternatively, the opticalaccelerometer 900 may include first and second reversing mirrorsdisposed in the second optical circuit 500′ instead of the beam combiner1100. The first and second reversing mirrors may be disposed at anglescomplementary to the first angle 700 (FIG. 7), relative to the lengthaxis 340 of the first retroreflector 407, to reflect the first andsecond beams 615 and 620, respectively. In this case, the beam splitter600 (FIGS. 6-9) may also act as a beam combiner, in a Sagnacconfiguration, as indicate by dashed lines in FIG. 11.

For illustrative purposes, the refractive index of each of the first andsecond optical media 325 and 325′ may be at least about 1.3, and therefractive index of each of the third and fourth optical media may be atmost about 1.002. As noted with respect to the embodiment described withreference to FIGS. 1-2, optical media with other indices of refractionmay be used, as long as the index of refraction of the first opticalmedium 325 is different from the index of refraction of the thirdoptical medium, and the index of refraction of the second optical medium325′ is different from the index of refraction of the fourth opticalmedium. Also, as noted, sensitivity (“gain”) depends in part on thedifference between the indices of refraction in the measurement cell andthe reference cell.

The first optical medium 325 may include glass. The second opticalmedium 325′ may include glass. The third optical medium may include afirst chamber containing a medium, such air, partial or complete vacuumor at least 99% nitrogen. The fourth optical medium may include a secondchamber containing a medium as selected for the first chamber.

As discussed in relation to the embodiment of FIGS. 1 and 2, the opticalaccelerometer 900 can operate in either a frequency-trackingaccelerometer mode, analogous to an IFOG, or in the integratedΔ-velocity mode, analogous to a PIGA. A good combination of features maybe obtained by operating in a combined mode, with anti-reflectioncoatings along most of the height of both sides of the measurement celloptical medium 320, operating in the integrated Δ-velocity mode forincreased sensitivity and a narrow band of high-reflection coatings onboth sides above the retroreflector 407′ at the beam combiner 1100 toconcentrate the first and second beams 615 and 620 to make their captureeasier.

FIG. 13 is a cut-away perspective schematic diagram of a toroidaloptical accelerometer 1300, according to another embodiment of thepresent invention. The toroidal optical accelerometer includes a specialcase interferometer. Essentially, the optical accelerometer 1300includes an element 300, as described with reference to FIGS. 3-8,wrapped about an input axis 1305. FIG. 15 is a perspective schematicdiagram of the element 300, similar to FIG. 6, indicating by extensionlines 1500 how the element 300 is curved to wrap it about the input axis1305. Returning to FIG. 13, the toroidal optical accelerometer 1300includes counter-facing first and second retroreflectors 1310 and 1315,as shown in more detail in FIG. 14. The first and second counterfacingtoroidal retroreflectors 1310 and 1315 have their toroidal centersdisposed on the input measurement axis 1305.

A toroidal first optical medium 1320 acts as a measurement cell andsurrounds a toroidal void 1325 filled with a second optical medium, suchas a partial or complete vacuum, air, dry nitrogen or another suitableoptical medium, which acts as a reference cell. An inside wall 1330 ofthe first optical medium 1320 and an outside wall of a tube 1335 definesides of the void 1325. Consequently, the second optical medium takes ona toroidal shape. The toroidal first optical medium 1320, the toroidalvoid 1325, the toroidal second optical medium and the tube 1335 arecoaxial about the input axis 1305. End caps, such as provided by thefirst and second retroreflectors 1310 and 1315, may define ends of thevoid 1325.

The first toroidal optical medium 1320 is disposed between the first andsecond counterfacing toroidal retroreflectors 1310 and 1315. The firsttoroidal optical medium 1320 has a first refractive index. The secondtoroidal optical medium has a second refractive index, different fromthe first refractive index. The second toroidal optical medium isdisposed between the first and second counterfacing toroidalretroreflectors 1310 and 1315.

A beam splitter 1340 splits an input light beam from a light source (notvisible) into two beams, as described with reference to FIG. 6. FIG. 16is a cut-away view of the optical accelerometer 1300, and FIG. 17 is across-sectional view of the optical accelerometer 1300 taken through thebeam splitter 1340. FIGS. 18 and 19 schematically illustrate paths takenby portions of the two beams 1900 and 1905. One beam 1905 initiallytraverses the first optical medium 1320, and the other beam 1900initially traverses the optical medium in the void 1325. When the beams1900 and 1905 emerge from the first and second optical media,respectively, they are reflected by the second retroreflector 1315, asdescribed with reference to FIG. 8. Thus, the two beams 1900 and 1905counter-rotate the optical circuit that includes the two optical media.The first beam 1900 traverses the circuit in a generally clockwisedirection, as indicated by arrow 1910, and the second beam 1905traverses the circuit in a generally counterclockwise direction.

The input light beam is introduced at an angle, other than zero,relative to the input axis 1305, as described with reference to FIG. 7.Therefore, the two light beams 1900 and 1905 traverse legs between thetwo retroreflectors 1310 and 1315 that are nearly, but not exactly,parallel to the input axis 1305. Consequentially, as the two beams 1900and 1905 reflect between the two retroreflectors 1310 and 1315, theyadvance circumferentially around the toroidal optical accelerometer1300, as indicated by an arrow 1345.

This advance is illustrated in FIG. 18, which shows only the second beam1905. For example, in a first leg 1800, the beam 1905 travels downthrough the void 1325 to the second retroreflector 1315, which reflectsthe beam 1905 so it travels in a second leg 1805 up through the firstoptical medium 1320. However, because the beam 1905 is not exactlyparallel to the input axis 1305, at the end of the first leg 1800, whenthe beam 1905 is reflected by the first half of the retroreflector 1315,the beam 1905 is reflected not exactly radially outward, as shown at1810. Consequently, the beam 1905 advances circumferentially, in thedirection of the arrow 1345. Each subsequent reflection by each of theretroreflectors 1310 and 1315, such as at 1815, 1820, 1825 and 1830,further advances the beam 1905 circumferentially, in the direction ofthe arrow 1345. Although not shown in FIG. 18, the other beam 1900 issimilarly reflected by the retroreflectors 1310 and 1315 to advancecircumferentially around the optical accelerometer 1300 in the directionof the arrow 1345. For purposes of illustration, the angles areexaggerated in FIG. 18.

In one embodiment, the beam splitter 1340 or a separate beam combiner(not shown) recombines the two beams 1900 and 1905, once the two beams1900 and 1905 have traveled around the circumference of the toroidaloptical accelerometer 1300. In another embodiment, as shown in FIG. 13,reversing mirrors 1350 and 1355 reflect the two beams 1900 and 1905,respectively, back along their paths to reverse the direction 1345 inwhich the two beams advance circumferentially around the toroidaloptical accelerometer 1300, and the beam splitter 1340 or a separatebeam combiner (not shown) recombines the two beams, once the two beamshave traveled back from the reversing mirrors 1350 and 1355.

An electronic circuit (not shown) may be disposed within the tube 1335or separately situated. The electronic circuit may be coupled to thelight source and to a light sensor (not shown) coupled to an output ofthe beam splitter 1340 or beam combiner, as the case may be. Theelectronic circuit may drive the light source, such as modulating thelight source as desired. The electronic circuit may analyze signals fromthe sensor to calculate acceleration along the input axis 1305experienced by the toroidal optical accelerometer 1300. The electroniccircuit may include a processor executing instructions, stored in amemory, to control the light source and/or to analyze the signals fromthe sensor and/or to calculate the acceleration.

The toroidal optical accelerometer 1300 is compact and inherentlyinsensitive to constant angular velocity Sagnac effects about the inputaxis 1305. If the two beams are reversed by the reversing mirrors 1350and 1355, the toroidal optical accelerometer 1300 is also insensitive toSagnac angular acceleration effects. Although the toroidal opticalaccelerometer 1300 is shown in FIGS. 13-19 as having the measurementcell radially outside the reference cell, these cells may beinterchanged. For example an outer tube (not shown) may define an outerboundary of the reference cell, and the outside surface of themeasurement cell may define an inner boundary of the reference cell.

FIG. 20 is a plan schematic diagram, and FIG. 21 is a left sideschematic diagram, of a planar waveguide optical accelerometer 2000having an input axis 2002, according to another embodiment of thepresent invention. The planar waveguide optical accelerometer includes aspecial case interferometer.

The optical accelerometer 2000 is constructed on a suitable substrate2004, such as Zerodur® glass. Alternating glass and vacuum, air or drynitrogen waveguides are deposited or constructed on the substrate 2004.Representative glass waveguides, which serve as measurement cells, areshown at 2006, 2008, 2010 and 2012. Representative vacuum waveguides,which serve as reference cells, are shown at 2014, 2016, 2018 and 2020.The vacuum waveguides may be defined by sides of the glass waveguides2006-2012, the substrate 2004 and a lid 2100, which may also be made ofthe same material as the substrate 2004. Reflective cladding may beapplied to surfaces of the waveguides 2006-2020 to constrain light beamswithin the waveguides and counteract beam divergence.

The substrate 2004 may be electromachined to define retroreflectors atends of each pair of waveguides, as exemplified by retroreflectors 2022and 2024. Optionally or alternatively, the retroreflectors 2022-2024 maybe implemented as photonic crystals. For example, two-dimensionalphotonic crystals may be made by photolithography, or by drilling holesin the substrate 2004, as shown in FIG. 22. Representative holes areindicated at 2200.

An input light beam 2026 is divided by a beam splitter 2028 into twolight beams 2030 and 2032 that traverse the planar waveguide opticalaccelerometer 2000 as indicated in FIG. 20. A beam combiner 2034combines the two beams 2030 and 2032 to produce an output light beam2036. As indicated by an arrow 2038, the first light beam 2030 traversesthe measurement/reference cells in a generally clockwise direction,whereas the second beam 2032 traverses the measurement/reference cellsin a generally counterclockwise direction, as indicated by an arrow2040.

In an example embodiment, a single about 4×6 cm device (dimensions 2042and 2044) is sufficient to provide a nominal phase-change read-outsensitivity of about 1 (rad/sec)/g, with losses of about 1 dB. In someembodiments, the substrate 2004 is about 500μ thick, the lid 2100 isabout 100μ thick and the waveguides 2006-2020 are each about 10μ high.The planar waveguide optical accelerometer 2000 may be fabricated usingconventional semiconductor MEMS fabrication techniques. For example,grooves may be etched in the substrate 2004. Half the grooves may befilled with optical glass, such as by vapor deposition, and theremaining grooves may be filled with dry gas or a partial vacuum.

Several substrates, each as described with respect to the substrate2004, may be layered together. Instead of a beam combiner 2034, eachlayer except the last layer includes a photonic coupler to animmediately adjacent layer. Only the last layer includes the beamsplitter 2028.

FIG. 23 is a perspective schematic illustration of a cylindricalwaveguide optical accelerometer 2300, according to an embodiment of thepresent invention. The cylindrical waveguide optical accelerometerincludes a special case interferometer. The cylindrical waveguideoptical accelerometer 2300 may be formed by wrapping the planarwaveguide optical accelerometer 2000, described with reference to FIGS.20-22, into a cylinder 2301 about an input axis 2302. Alternatingmeasurement cells and reference cells, represented by measurement cells2304 and 2306 and reference cells 2308 and 2310, are disposed parallelto the input axis 2302 and about an outer surface of a substrate 2312.The substrate 2312 may be any suitable material, such as Zerodur® glass.

A cylindrical lid (not shown for clarity), which may be made of the samematerial as the substrate 2312 or of a different suitable material,covers the measurement and reference cells 2304-2310. retroreflectors,represented by retroreflectors 2314 and 2316, are disposed at ends ofpairs of the cells 2304-2310, as discussed with reference to FIG. 20. Abeam splitter 2318 and a beam combiner 2320 are disposed are respectiveends of two of the cells.

As with the toroidal optical accelerometer 1300 described with referenceto FIGS. 13-19, an electronic circuit (not visible) may be disposedwithin a space 2321 defined by the cylinder 2301 or separately situated.The electronic circuit may be coupled to a light source and to a lightsensor (not shown) respectively coupled to the beam splitter 2318 and tothe beam combiner 2320. The electronic circuit may drive the lightsource, such as modulating the light source as desired. The electroniccircuit may analyze signals from the sensor to calculate accelerationalong the input axis 2302 experienced by the cylindrical waveguideoptical accelerometer 2300.

As shown in FIG. 24, several cylinders, represented by cylinders 2400,2402 and 2404, each similar to the cylinder 2301 (FIG. 23), may benested coaxially about an input axis 2406. The outer cylinder 2404 maybe covered by a cylindrical lid 2408. The nested cylinders 2400-2404 areshown exploded along the input axis 2406 in FIG. 24 and collapsed into acompact cylindrical form in FIG. 25.

Each cylinder 2402-2404 defines a light path that traverses themeasurement and reference cells of the cylinder. The light paths ofintermediate cylinders 2402-2404 are optically coupled to light paths inimmediately adjacent cylinders to form a single optical path extendingserially from the beginning of outer most cylinder 2404 (not countingthe lid 2408) to the end of the inner most cylinder 2400. This longoptical path provides increased gain over that available from a singlecylinder.

As with the cylindrical waveguide optical accelerometer 2300 describedwith reference to FIG. 24, an electronic circuit (not visible) may bedisposed within a space 2410 defined by the inner most cylinder 2400 orseparately situated. The electronic circuit may be coupled to a lightsource and to a light sensor (not shown) respectively coupled to a beamsplitter in the outer most cylinder 2404 (alternatively, the inner mostcylinder 2400) and to a beam combiner in the inner most cylinder 2400(alternatively, the outer most cylinder 2404). The electronic circuitmay drive the light source, such as modulating the light source asdesired. The electronic circuit may analyze signals from the sensor tocalculate acceleration along the input axis 2406 experienced by thenested cylindrical waveguide optical accelerometer.

FIG. 26 is a perspective schematic illustration of a spiral-woundwaveguide optical accelerometer 2600, according to another embodiment ofthe present invention. The optical accelerometer 2600 is similar to theoptical accelerometer 2300, except a single substrate 2602 is spiralwound and disposed within a cylindrical lid 2604. Advantages of thespiral-wound optical accelerator 2600 include obviating a need tooptically couple a plurality of coaxial substrates.

As described herein, various embodiments of optical accelerometersinclude measurement cells and reference cells, in which the measurementcells have different indices of refraction than the reference cells, andoptical paths are defined such that two light beams traverse themeasurement and reference cells in opposite directions. FIG. 27 is aperspective schematic illustration of a fiber optic accelerometer 2700according to yet another embodiment of the present invention. The fiberoptic accelerometer includes a special case interferometer.

An optical fiber 2702 is wound much like a diagonally-wound ball oftwine. The ball is wound circumferentially about an axis 2704. Theoptical fiber 2702 forms a coil and conducts a light beam 2703. Theoptical fiber 2702 is prepared ahead of time, or while the ball iswound, so as to form half-winding pairs of the optical fiber 2702, suchas half-winding pair 2706 and 2708 and half-winding pair 2710 and 2712,where the light beam 2703 is conducted by one half-winding of each pairand then immediately by the other half-winding of the pair. Eachhalf-winding of each pair conducts the light beam 2703 in a directionopposite that of the other half-winding of the pair, relative to aninput axis 2713.

One half-winding of each pair has a different index of refraction thanthe other half-winding of the pair. One half-winding, indicated by asolid line, of each pair forms a measurement cell, and the otherhalf-winding, indicated by a dashed line, of the pair forms a referencecell. A winding direction reversal mid-way through the optical fibercoil winding, indicated as 2714, so that one half of the coil may bewound in a generally clockwise direction about the axis 2713, while theother half may be wound generally counter-clockwise about the axis 2713,as shown by the two curved arrows around axis 2713, provides a zero-areaSagnac rotation compensation.

Such an optical fiber 2702 may be made of double-clad optical fiber thatis irradiated by a laser 2718 to form double-clad long-period fiberBragg gratings (DLFBG), represented by DLFBG 2720, to change the indexof refraction of the fiber along its length, within the extent of theDLFBG, on alternate half-windings of the optical fiber, in particular,on the measurement cells. To form the DLFBG, the irradiation may includeone wavelength or a plurality of wavelengths. The irradiation intensitymay be constant or varied. The irradiation may be performed using one ora plurality of lasers. The laser may be a focused pulsed laser. Theoptical fiber may be irradiated, to form the DLFBG, while the opticalfiber is being wound. Also, the irradiation of optical fiber may beperformed separately from winding the optical fiber, in any order. Aportion 2722 of the optical fiber 2702 is shown in FIG. 27. A length2724 of the optical fiber 2702 equals approximately one-half thecircumference (one-half winding) of the ball. Note that thecircumference of the ball increases as the ball is wound.

The length 2724 of the optical fiber 2702 is treated, at one-half thecircumference intervals, by the laser 2718 to form the DLFBG to form themeasurement cells. All the measurement cells need not be the samelength. Adjacent lengths 2726 and 2728, which are each approximatelyone-half the circumference long, are untreated by the laser 2718,forming respective reference cells.

FIG. 28 schematically illustrates the same portion 2722 of the opticalfiber 2702, as well as schematically illustrating four cross-sections ofthe optical fiber 2702. Section A shows an air core 2800 and cladding2802 of the optical fiber 2702. As shown in Section B, in the referencecell 2726, the light 2804 is conducted along the air core 2800. However,as shown in Section C, the DLFBG of the measurement cell causes thelight 2806 to switch from the air core 2800 to the cladding 2802, asindicated by arrows at 2808. As shown in Section D, once the light exitsthe measurement cell 2724, the light 2810 switches back to the air core2800, as indicated by arrows 2812.

The cladding 2802 has a higher index of refraction than the air core2800. Therefore, as the light beam 2703 traverses the measurement cells,the light beam experiences a higher index of refraction than when thelight beam traverses the reference cells. The portions of the opticalfiber 2702 that have the first index of refraction are contiguous withsubsequent adjacent portions of the optical fiber with the second indexof refraction. Similarly, the portions of the optical fiber 2702 thathave the second index of refraction are contiguous with subsequentadjacent portions of the optical fiber with the first index ofrefraction, i.e., there are no splices between adjacent portions withdifferent indices of refraction. Other aspects of the fiber opticaccelerometer 2700 operate according to the principles discussed herein.

Instrument describing equations for phase shift and frequency shift arepresented for two independent methods: by integration of the phase shiftaround the beam path; and by use of the relativistic velocity additionformula. Dispersion terms have been neglected, as monochromatic laserillumination is assumed, and dispersion effects will cancel out, as hasbeen shown by [13]. Both methods give identical results, which areconsistent with published results of studies of other related devices.Exemplary calculations follow.

In 1967, Post [1] calculated the phase integral of the electromagneticfields of a laser gyro's light beams about the gyro's closed path byintegration by parts, applies Stokes's theorem to equate this to therotational flux through the path's enclosed area, and then demonstratedthe method on gyro and interferometer beams imbedded in static andconstant velocity media. Post [1], Eq. (46), gives the phase shift, infractional wavelengths, for a contra-rotating beam Sagnac interferometerand refractive medium moving together at a uniform velocity, as:

$\begin{matrix}{{{\Delta\; Z} = {\frac{2}{C\;\lambda_{0}}{\oint{{n^{2}\left( {1 - \alpha} \right)}{\overset{\_}{V} \cdot \overset{\_}{ds}}}}}},{{{where}\mspace{14mu}\alpha} = \left( {1\text{-}{1/n^{2}}} \right)}} & (1)\end{matrix}$When coupled with the accessory condition that δω=0, this leads to:

$\begin{matrix}{{\Delta\; Z} = {\frac{2}{C\;\lambda_{0}}{\oint{\overset{\_}{V} \cdot \overset{\_}{ds}}}}} & (2)\end{matrix}$and the well-known conclusion [13] that the phase shift in a Sagnacinterferometer in constant rotation, to first order, is independent ofthe characteristics of the medium, dependent only upon the area of theclosed path and the rotational field through it, and is unaffected byuniform translation.

Under acceleration, however, with separate beams passing throughdifferent media, the conditions are different. Once a light beam islaunched from the source, the photon phase velocity in vacuum isindependent of the source velocity, and is constant at 3×10⁸ m/sec,while the photon frequency is Doppler shifted by the moving source,whether accelerating or at constant velocity. At constant velocity, thisis compensated for by an identical and opposite shift at the detector.Under acceleration, the integrated acceleration of the detector duringthe photon's transit to the detector results in a differential velocitybetween source and detector, causing an additional Doppler effect at thedetector. However, in a two beam Mach-Zehnder interferometer these willbe identical in the reference and measurement beams and will thus alsobe compensated for. The Doppler effects of differential velocity due toacceleration of each medium's entering & leaving interfaces are secondorder, and can be absorbed into the device scale factor.

What remains is the change in interaction between the constant vacuumvelocity photon beams with the atoms of the measurement and referencecells, due to the change in kinetic energy and momentum of the atomsunder acceleration, resulting in additional frequency and integratedphase shifts in each beam. If the indices of refraction of the mediumsthe beams pass through are equal, there will be no net interferometeroutput, but if they are unequal, there will be a net change in theoutput, proportional to the acceleration. (In a transparent medium, themomentum and energy transfer from the photon beam to the medium is verysmall (the definition of “transparent”), relative to the momentum andenergy transfer from the moving medium to the photon beam.)

As this is the only remaining effect of acceleration on this device, thefrequency and phase shift for an accelerating medium in an acceleratingMach-Zehnder interferometer in vacuum will be the same as for anaccelerating medium in a static Mach-Zehnder interferometer in vacuum,and the discussion of Post [1], Section IV-D and equation [1-(55)] areappropriate, with velocity, V, replaced with the integratedacceleration, ΔV, The phase shift becomes:

${\Delta\; Z} = {\frac{2}{C\;\lambda_{0}}{\oint{\left( {n^{2}\alpha} \right)\Delta{\overset{\_}{V} \cdot \overset{\_}{ds}}}}}$which is equivalent to:

$\begin{matrix}{{\Delta\; Z} = {\frac{2}{C\;\lambda_{0}}{\oint{\left( {1 - n^{2}} \right)\Delta{\overset{\_}{V} \cdot \overset{\_}{ds}}}}}} & (3)\end{matrix}$

For the interferometer with figure-8 loops of equal loops of FIGS. 1 and2, accelerating from one uniform velocity field, V₁, (|V₁|<<C), toanother, V₂, (|V₂|<<C), where ΔV=(V₂−V₁)·IA, the path length can bebroken into three parts:

1. The two loops, imbedded in media of equal and uniform indices ofrefraction, excluding the measurement and reference cells. The pathlength in the forward and backward velocity directions are equal, so,excluding the possibility of non-reciprocal effects due to birefringenceor polarization, the path integral of these segments and theircontribution to the resultant phase shift will be 0.

2. The phase shift contribution due to the total path integral forcounter-rotating beams through both of the measurement cells, assumingtheir index of refraction (n_(m)) is uniform and equal, and they are ofequal length (L), is:

$\begin{matrix}{{\Delta\; Z_{m}} = {\frac{2}{C\;\lambda_{0}}\left( {2{L\left( {1 - n_{m}^{2}} \right)}\Delta\; V} \right.}} & (4)\end{matrix}$

3. Similarly, because the acceleration direction through both thereference cells is reversed for a translating interferometer relative tothe beam direction for a rotating interferometer, their phase shiftcontribution, again assuming their index of refraction (n_(r)) isuniform and equal, and they are of equal length to the measurement cells(L), is:

$\begin{matrix}{{\Delta\; Z_{r}} = {\frac{- 2}{C\;\lambda_{0}}\left( {2\;{L\left( {1 - n_{r}^{2}} \right)}\Delta\; V} \right.}} & (5)\end{matrix}$

If n_(r) is the same as that of the underlying structure excluding themeasurement and reference cells (e.g., n_(r)=1, vacuum) the length ofthe reference cell will automatically be the same as that of themeasurement cell without the need for partitioning off a separate cell.

The sum of the phase shift contributions is thus:

$\begin{matrix}{{{\Delta\; Z} = {\frac{4\; L}{C\;\lambda_{0}}\left( {n_{r}^{2} - n_{m}^{2}} \right)\Delta\; V}},\left\{ {{wavelengths},\left( {{with}{\mspace{11mu}\;}\Delta\mspace{14mu} V\mspace{14mu}{in}\mspace{14mu} m\text{/}\sec} \right)} \right\}} & (6)\end{matrix}$

Note that if the reference cells are vacuum (if the reference cellmedium is vacuum, that leg, as well as the underlying structure isunaffected by acceleration (see also Footnote 5)), this reduces to:

$\begin{matrix}{{{\Delta\; Z_{0}} = {\frac{4\; L}{C\;\lambda_{0}}\left( {1 - n_{m}^{2}} \right)\Delta\; V}},\left\{ {{wavelengths},\left( {{with}{\mspace{11mu}\;}\Delta\mspace{14mu} V\mspace{14mu}{in}\mspace{14mu} m\text{/}\sec} \right)} \right\}} & (7)\end{matrix}$Or, in scale factor form:

$\begin{matrix}{{{{\frac{\Delta\; Z}{\Delta\; V} = {\frac{4L}{C\;\lambda_{0}}\left( {n_{r}^{2} - n_{m}^{2}} \right)}},\mspace{14mu}{and}}\quad}\mspace{20mu}{\frac{\Delta\; Z_{0}}{\Delta\; V} = {\frac{4\; L}{C\;\lambda_{0}}\left( {1 - n_{m}^{2}} \right)\mspace{14mu}\left\{ {{wavelengths},\left( {{with}{\mspace{11mu}\;}\Delta\mspace{14mu} V\mspace{14mu}{in}\mspace{14mu} m\text{/}\sec} \right)} \right\}}}} & (8)\end{matrix}$

The equivalent expression for a single pass through a single layer ofthe vertical slab interferometer with counter-rotating beams of FIG. 3,as shown in FIG. 8, is:

$\begin{matrix}{{{{\frac{\Delta\; Z}{\Delta\; V} = {\frac{2L}{C\;\lambda_{0}}\left( {n_{r}^{2} - n_{m}^{2}} \right)}},\mspace{14mu}{and}}\quad}{\;\mspace{14mu}}{\frac{\Delta\; Z_{0}}{\Delta\; V} = {\frac{2\; L}{C\;\lambda_{0}}\left( {1 - n_{m}^{2}} \right)\mspace{14mu}\left\{ {{wavelengths},\left( {{with}{\mspace{11mu}\;}\Delta\mspace{14mu} V\mspace{14mu}{in}\mspace{14mu} m\text{/}\sec} \right)} \right\}}}} & \left( {8a} \right)\end{matrix}$

Consider a single pass through one layer of the Mach-Zehnderinterferometer of FIGS. 6-8, excited by a fixed frequency source,f₀=ω₀/2π, at rest in a local inertial frame, V₀, with index ofrefraction of the reference cell, n_(r), and of the measurement cell,n_(m).

The two paths followed by the first and second beams 120 and 125 areidentical, but reversed, and are imbedded in media of equal and uniformindices of refraction, excluding the measurement and reference cells.The path length of the two beams are nominally equal so, excluding thepossibility of second-order non-reciprocal effects due to birefringenceor polarization, their contributions to the resultant phase shiftmeasured at the beam combiner 1100, other than that due to the differingindices of refraction of the measurement and reference cells, willnominally be zero.

The speed of light through the reference cells is u_(r)=C/n_(r), andthrough the measurement cells, u_(m)=C/n_(m). The corresponding transittimes through cells of length, L are τ_(r)=L(n_(r)/C) andτ_(m)=L(n_(m)/C).

The phase shift across each cell, respectively, is then ϕ_(r)=f₀ τ_(r)and ϕ_(m)=f₀ τ_(m) and the phase shift across the interferometer is: thedifference of the sums of the phase shifts of the measurement cells 135and 150 and reference cells 140 and 160.

$\begin{matrix}{{\Delta\varphi} = {{f_{0}\left( {\tau_{r} - \tau_{m}} \right)} = {\frac{2\; L}{\lambda_{0}}\left( {n_{r} - n_{m}} \right)}}} & (9)\end{matrix}$

This is the phase shift in an unbalanced Mach-Zehnder interferometer dueto dielectric cells of different index of refraction in a uniforminertial reference frame at constant, (arbitrary) velocity. As above,this phase shift is either mechanically adjusted to 0, or measured andrecorded as a reference for the future.

A useful equivalent form is:

$\begin{matrix}{{{{\Delta\varphi}\left( \frac{{C \cdot 2}\; L}{\lambda_{0}} \right)}\left( {\frac{n_{r}}{C} - \frac{n_{m}}{C}} \right)} = {{\frac{{C \cdot 2}\; L}{\lambda_{0}}\left( {\frac{1}{u_{r}} - \frac{1}{u_{m}}} \right){\Delta\varphi}\frac{{C \cdot 2}\; L}{\lambda_{0}}\left( {\frac{n_{r}}{C} - \frac{n_{m}}{C}} \right)} = {\frac{{C \cdot 2}\; L}{\lambda_{0}}\left( {\frac{1}{u_{r}} - \frac{1}{u_{m}}} \right)}}} & (10)\end{matrix}$

The interferometer is now uniformly accelerated parallel to the InputAxis over a period of time to a new velocity, V₁, where ΔV=V₁−V₀.Through relativistic velocity addition [14-(53)], the velocity of lightwithin the reference cell medium, with parameters as measured in theoriginal reference frame [15], becomes:

$\begin{matrix}{u_{r}^{\prime} = {\frac{{dx}_{r}^{\prime}}{{dt}^{\prime}} = {\frac{u_{r} - {\Delta\; V}}{1 - \frac{\Delta\;{V \cdot u_{r}}}{C^{2}}} = {\frac{\frac{C}{n_{r}} - {\Delta\; V}}{1 - {\frac{\Delta\; V}{C^{2}} \cdot \frac{C}{n_{r}}}} = \frac{\frac{C}{n_{r}} \cdot \left( {1 - \frac{\Delta\;{V \cdot n_{r}}}{C}} \right)}{1 - \frac{\Delta\; V}{n_{r} \cdot C}}}}}} & (11) \\{\frac{1}{u_{r}^{\prime}} = {{\frac{n_{r}}{C} \cdot \left( {1 - \frac{\Delta\; V}{n_{r} \cdot C}} \right) \cdot \left( {1 - \frac{\Delta\;{Vn}_{r}}{C}} \right)^{- 1}} = {\frac{n_{r}}{C} \cdot \left( {1 - \frac{\Delta\; V}{n_{r} \cdot C}} \right) \cdot \left( {1 + \frac{\Delta\;{Vn}_{r}}{C} + \left( \frac{\Delta\;{Vn}_{r}}{C} \right)^{2} + {.\;.\;.}}\mspace{14mu} \right)}}} & (12)\end{matrix}$Neglecting higher order terms than ΔV/C²:

$\begin{matrix}{\frac{1}{u_{r}^{\prime}} = {{\frac{n_{r}}{C} \cdot \left( {1 - \frac{\Delta\; V}{n_{r} \cdot C} + \frac{\Delta\;{V \cdot n_{r}}}{C}} \right)} = {\frac{n_{r}}{C} + {\frac{\Delta\; V}{C^{2}}\left( {n_{r}^{2} - 1} \right)}}}} & (13)\end{matrix}$Similarly:

$\begin{matrix}{\frac{1}{u_{m}^{\prime}} = {\frac{n_{m}}{C} + {\frac{\Delta\; V}{C^{2}}\left( {n_{m}^{2} - 1} \right)}}} & (14)\end{matrix}$

Inserting (13) and (14) into (10) gives:

$\begin{matrix}{{\Delta\varphi} = {{\frac{{C \cdot 2}\; L}{\lambda_{0}} \cdot \left( {\frac{n_{r} - n_{m}}{C} + {\frac{\Delta\; V}{C^{2}} \cdot \left( {n_{r}^{2} - n_{m}^{2}} \right)}} \right)} = {{\underset{\underset{{Static},{= \varphi_{0}}}{︸}}{\frac{2\; L}{\lambda_{0}} \cdot \left( {n_{r} - n_{m}} \right)} + \underset{\underset{{Dynamic},{= {({\varphi_{1} - \varphi_{0}})}}}{︸}}{\frac{\Delta\;{V \cdot 2}\; L}{\lambda_{0} \cdot C} \cdot \left( {n_{r}^{2} - n_{m}^{2}} \right)}} = {\Delta\; Z_{0}}}}} & (15)\end{matrix}$

The first (“Static”) term in Eq. (15) is identical to Eq. (9), theinterferometer's phase shift at V₀, the original reference velocity, andthe second (“Dynamic”) term is the additional phase shift in theaccelerated frame due to the change in velocity ΔV (the Fizeau dragterm), identical in form to Eq. (6). This second term is the output of asplit-beam, single layer, figure-8 configured, Mach-Zehnder integratingaccelerometer with a non-vacuum Reference Cell that has been adjustedfor 0 phase shift in the reference (launch) frame.

From Eq. (13), the index of refraction and velocity of propagation inthe accelerated frame, to the first order in ΔV, is:

$\begin{matrix}{{\frac{C}{u_{V\; 1}} = {n_{V\; 1} = \left( {n_{V\; 0} + {\left( {n_{V\; 0}^{2} - 1} \right) \cdot \left( \frac{\Delta\; V}{C} \right)}} \right)}}{and}} & (16) \\{u_{V\; 1} = \left( {u_{V\; 0} - {\left( {1 - \frac{1}{n_{V\; 0}^{2}}} \right) \cdot \left( {\Delta\; V} \right)}} \right)} & (17)\end{matrix}$

Through a rearrangement of terms, (16) can be shown to be identical toequation (13) of [16], for a reversal of direction, V=−ΔV:

$\begin{matrix}{n^{\prime} = \frac{n + \left( {V/C} \right)}{{1 + {n\left( {V/C} \right)}}}} & \left\lbrack {16\text{-}(13)} \right\rbrack\end{matrix}$

Assume a pair of contra-rotating beams passing through a single layer ofthe layered-loop vertical slab interferometer of FIG. 3 with thefollowing nominal parameter values:

L=2 inches=5E-2 m, C=3E8 m/sec, λ₀=1.55E-6 m, n_(m)=1.46 (Infrasil 301optical glass), n_(r)=1 (vacuum). In this case, there is one measurementcell and reference cell, so the scale factor in the integratedΔ-velocity mode is:

$\begin{matrix}{{S.F._{({\Delta\; V})}} = {\frac{\Delta\; Z_{0}}{\Delta\; V} = {{\frac{2\; L}{\lambda_{0}C}\left( {1 - n_{m}^{2}} \right)} = {{\frac{2\; x\;{5 \cdot 10^{- 2}}}{{1.55 \cdot 10^{- 6}}\; x\;{3 \cdot 10^{8}}}\left( {1 - 2.13} \right)} = {{- 2.43} \cdot 10^{- 4}}}}}} & \left( {{37a},b} \right)\end{matrix}$

-   -   {Wavelengths/(m/sec), or 1.53 mrad/(m/sec ΔV), per layer}

For a bulk optics instrument with contra-rotating beams, a 0.5 deg. (8mrad) tilted entrance beam gives an 0.08 cm rise of successive beamlayers over a 2×5 cm path length, for 12.5 passes/cm, or 19 mrad/(m/secΔV) per cm height. (Consistent with currently available laser sourcefree-space beam divergence. Greater divergence may require acorrespondingly greater rise between successive beams, correctivefiguring to the surfaces of the measurement cell, or a measurement cellcomposed of planar wave guides, as in FIG. 20, aligned fiber-opticbundles, or ruled gratings, to capture and control the light beam.)

To provide a nominal read-out sensitivity of 1 (rad/sec)/g, equivalentto 0.1 (rad)/(m/sec ΔV), with Sagnac effect immunity for a device withthe example dimensions requires about 66 layers, or a stacked pair ofinstrument blocks about 2.7 cm high, as in FIG. 12. Different dimensionswill provide different performance.

As another example, the concentric toroidal configuration of FIGS.13-19, with tilted input beams to provide multiple passes for increasedgain, has an integrated Δ-velocity mode scale factor of:

$\begin{matrix}{\frac{\Delta\; Z_{T}}{\Delta\; V} = {\frac{4\pi\; L}{C\;\lambda_{0}}\left( \frac{\pi\; D}{s} \right)\left( {n_{r}^{2} - n_{m}^{2}} \right)\mspace{14mu}{rad}\text{/}\left( {m\text{/}\sec} \right)}} & (38)\end{matrix}$where πD/s is the number of turns around the toroid perimeter and s isthe separation between spiral loops.

Using the same 0.08 cm separation (s) between spiral loops as above, anominal 4.4 cm diameter (D) of the toroid, and a nominal 5 cm celllength (L), provides 170 spiral passes for a single circuit around thetoroid, for a one-way Δ-velocity mode instrument scale factor of 260mrad/(m/sec ΔV).

While specific parameter values may be recited for disclosedembodiments, within the scope of the invention, the values of all ofparameters may vary over wide ranges to suit different applications.While the invention is described through the above-described exemplaryembodiments, modifications to, and variations of, the illustratedembodiments may be made without departing from the inventive conceptsdisclosed herein. Furthermore, disclosed aspects, or portions thereof,may be combined in ways not listed above and/or not explicitly claimed.Accordingly, the invention should not be viewed as being limited to thedisclosed embodiments.

Although aspects of embodiments may be described with reference toflowcharts and/or block diagrams, functions, operations, decisions, etc.of all or a portion of each block, or a combination of blocks, may becombined, separated into separate operations or performed in otherorders. Equations may be rearranged to generate equivalent results. Allor a portion of each block, or a combination of blocks, may beimplemented as computer program instructions (such as software),hardware (such as combinatorial logic, Application Specific IntegratedCircuits (ASICs), Field-Programmable Gate Arrays (FPGAs) or otherhardware), firmware or combinations thereof. Embodiments may beimplemented by a processor executing, or controlled by, instructionsstored in a memory. The memory may be random access memory (RAM),read-only memory (ROM), flash memory or any other memory, or combinationthereof, suitable for storing control software or other instructions anddata. Instructions defining the functions of the present invention maybe delivered to a processor in many forms, including, but not limitedto, information permanently stored on tangible non-writablenon-transitory storage media (e.g., read-only memory devices within acomputer, such as ROM, or devices readable by a computer I/O attachment,such as CD-ROM or DVD disks), information alterably stored on tangiblewritable non-transitory storage media (e.g., floppy disks, removableflash memory and hard drives) or information conveyed to a computerthrough a communication medium, including wired or wireless computernetworks. Moreover, while embodiments may be described in connectionwith various illustrative data structures, systems may be embodied usinga variety of data structures.

Changes may be made in the above methods and systems without departingfrom the scope hereof. It should thus be noted that the matter containedin the above description or shown in the accompanying drawings should beinterpreted as illustrative and not in a limiting sense. The followingclaims are intended to cover all generic and specific features describedherein, as well as all statements of the scope of the present method andsystem, which, as a matter of language, might be said to falltherebetween.

What is claimed is:
 1. A waveguide optical accelerometer, comprising: asubstrate; first and second retroreflector modules counterfacing along ameasurement axis, disposed on the substrate, and cooperating to steer afirst optical beam and a second optical beam along respective first andsecond optical paths of opposite helicities, each of the first andsecond optical paths forming an alternating sequence of parallel passesand antiparallel passes between the retroreflector modules, each of theparallel passes oriented parallel to the measurement axis and each ofthe antiparallel passes oriented antiparallel to the measurement axis;and a plurality of first optical waveguides formed from a first opticalmedium having a first refractive index, the first optical waveguidesdisposed on the substrate between the retroreflector modules such thatthe first optical beam propagates through one of the first opticalwaveguides for each of the parallel passes and the second optical beampropagates through one of the first optical waveguides for each of theantiparallel passes; a lid disposed over the first optical waveguidesand retroreflector modules, the lid cooperating with the substrate andpairs of adjacent first optical waveguides to define a plurality ofsecond optical waveguides, formed from a second optical medium having asecond refractive index different from the first refractive index, suchthat the second optical beam propagates through one of the secondoptical waveguides for each of the parallel passes and the first opticalbeam propagates through one of the second optical waveguides for each ofthe antiparallel passes; wherein the first optical medium and the secondoptical medium cooperate to differentially phase-shift the first opticalbeam and the second optical beam according to an acceleration of thewaveguide optical accelerometer along the measurement axis.
 2. Thewaveguide optical accelerometer of claim 1, further comprising abeamsplitter disposed on the substrate and configured to (i) divide aninput optical beam into the first optical beam and the second opticalbeam, and (ii) steer the first optical beam and the second optical beaminto the first and second optical paths, respectively.
 3. The waveguideoptical accelerometer of claim 1, further comprising a beam combinerconfigured to combine the first optical beam and the second optical beamafter propagating along the first and second optical paths,respectively.
 4. The waveguide optical accelerometer of claim 1, each ofthe first optical waveguides having a reflective cladding to constraineither one of the first optical beam and the second optical beampropagating therein.
 5. The waveguide optical accelerometer of claim 1,the lid and the substrate being formed of the same material.
 6. Thewaveguide optical accelerometer of claim 1, at least one of theretroreflector modules including photonic crystals.
 7. The waveguideoptical accelerometer of claim 1, the first optical medium being glass;and the second optical medium being one of vacuum, air, and drynitrogen.
 8. The waveguide optical accelerometer of claim 1, thesubstrate being a planar substrate oriented such that the alternatingsequence of parallel passes and antiparallel passes advancesperpendicularly to the measurement axis.
 9. The waveguide opticalaccelerometer of claim 1, the substrate being a cylindrical substratewrapped around the measurement axis such that the alternating sequenceof parallel passes and antiparallel passes advances circumferentiallyaround the measurement axis.
 10. The waveguide optical accelerometer ofclaim 1, the substrate being a spiral substrate wrapped around themeasurement axis such that the alternating sequence of parallel passesand antiparallel passes advances angularly around the measurement axis.11. A stacked-waveguide optical accelerometer, comprising: a firstinstance of the waveguide optical accelerometer of claim 8; a secondinstance of the waveguide optical accelerometer of claim 8 stacked ontop of the first instance such that the measurement axis of the secondinstance and the measurement axis of the first instance are parallel;the first instance of the waveguide optical accelerometer furtherincluding: a beamsplitter disposed on the substrate and configured to(i) divide an input optical beam into the first optical beam and thesecond optical beam, and (ii) steer the first optical beam and thesecond optical beam into the respective first and second optical pathsof the first instance; and a photonic output coupler configured tocouple the first and second optical beams, after propagating along therespective first and second optical paths of the first instance, fromthe first device to the second instance; and the second instance of thewaveguide optical accelerometer further including: a photonic inputcoupler configured to receive the first and second optical beams fromthe photonic output coupler of the first instance; and a beam combinerdisposed on the substrate and configured to combine the first opticalbeam and the second optical beam after propagating along the respectivefirst and second optical paths of the second instance.
 12. Thestacked-waveguide optical accelerometer of claim 11, the lid and thesubstrate of the first instance of the waveguide optical accelerometerbeing formed of the same material; and the lid and the substrate of thesecond instance of the waveguide optical accelerometer being formed ofthe same material.
 13. The stacked-waveguide optical accelerometer ofclaim 11, the substrate of the second instance of the waveguide opticalaccelerometer being the lid of the first instance of the waveguideoptical accelerometer.
 14. The stacked-waveguide optical accelerometerof claim 11, each of the first optical waveguides of having a reflectivecladding to constrain either one of the first optical beam and thesecond optical beam propagating therein.
 15. The stacked-waveguideoptical accelerometer of claim 11, the first optical medium being glass;and the second optical medium being one of vacuum, air, and drynitrogen.
 16. A coaxial nested-waveguide optical accelerometer,comprising: a first instance of the waveguide optical accelerometer ofclaim 9 having a first cylindrical diameter; a second instance of thewaveguide optical accelerometer of claim 9 having a second cylindricaldiameter less than the first cylindrical diameter, the second instancenested inside the first instance such that the measurement axis of thefirst instance and the measurement axis of the second instance coincide;the first instance of the waveguide optical accelerometer furtherincluding: a beamsplitter disposed on the substrate and configured to(i) divide an input optical beam into the first optical beam and thesecond optical beam, and (ii) steer the first optical beam and thesecond optical beam into the respective first and second optical pathsof the first instance; and a photonic output coupler configured tocouple the first and second optical beams, after propagating along therespective first and second optical paths of the first instance, fromthe first instance to the second instance; the second instance of thewaveguide optical accelerometer further including: a photonic inputcoupler configured to receive the first and second optical beams fromthe photonic output coupler of the first instance; and a beam combinerdisposed on the substrate and configured to combine the first opticalbeam and the second optical beam after propagating along the respectivefirst and second optical paths of the second instance.
 17. The coaxialnested-waveguide optical accelerometer of claim 16, the lid and thesubstrate of the first instance of the waveguide optical accelerometerbeing formed of the same material; and the lid and the substrate of thesecond instance of the waveguide optical accelerometer being formed ofthe same material.
 18. The coaxial nested-waveguide opticalaccelerometer of claim 16, the substrate of the second instance of thewaveguide optical accelerometer being the lid of the first instance ofthe waveguide optical accelerometer.
 19. The coaxial nested-waveguideoptical accelerometer of claim 16, each of the first optical waveguideshaving a reflective cladding to constrain either one of the firstoptical beam and the second optical beam propagating therein.
 20. Thecoaxial nested-waveguide optical accelerometer of claim 16, the firstoptical medium being glass; and the second optical medium being one ofvacuum, air, and dry nitrogen.